Tightness and convergence of trimmed Levy process to normality at small times

Abstract

For nonnegative integers r, s, let (r,s)Xt(r,s)Xt be the Lévy process XtXt with the r largest positive jumps and the s smallest negative jumps up till time t deleted, and let (r)X˜t(r)X~t be XtXt with the r largest jumps in modulus up till time t deleted. Let at∈Rat∈R and bt>0bt>0 be non-stochastic functions in t. We show that the tightness of ((r,s)Xt−at)/bt((r,s)Xt−at)/bt or ((r)X˜t−at)/bt((r)X~t−at)/bt as t↓0t↓0 implies the tightness of all normed ordered jumps, and hence the tightness of the untrimmed process (Xt−at)/bt(Xt−at)/bt at 0. We use this to deduce that the trimmed process ((r,s)Xt−at)/bt((r,s)Xt−at)/bt or ((r)X˜t−at)/bt((r)X~t−at)/bt converges to N(0, 1) or to a degenerate distribution as t↓0t↓0 if and only if (Xt−at)/bt(Xt−at)/bt converges to N(0, 1) or to the same degenerate distribution, as t↓0t↓0 Show more